CAT 2023 Slot 3 QA Question 22: suppose $f(x,y)$ is a real-valued function such that $f(3x + 2y, 2x - 5y) = 19x$, for all real numbers x and y. The value of x for which $f(x, 2x) = 27$, is

Question

suppose $f(x,y)$ is a real-valued function such that $f(3x + 2y, 2x - 5y) = 19x$, for all real numbers x and y. The value of x for which $f(x, 2x) = 27$, is

Accepted Answer

Use different variables in the given function relation to avoid confusion. Let

$f(3u + 2v, 2u - 5v) = 19u$

We need $f(x, 2x) = 27$. So match the two arguments:

$3u + 2v = x$
$2u - 5v = 2x$

Substitute $x = 3u + 2v$ in the second equation.

$2u - 5v = 2(3u + 2v)$
$2u - 5v = 6u + 4v$
$4u + 9v = 0$

suppose f(x,y)f(x,y)f(x,y) is a real-valued function such that f(3x+2y,2x−5y)=19xf(3x + 2y, 2x - 5y) = 19xf(3x+2y,2x−5y)=19x, for all real numbers x and y. The value of x for which f(x,2x)=27f(x, 2x) = 27f(x,2x)=27, is

suppose $f(x,y)$ is a real-valued function such that $f(3x + 2y, 2x - 5y) = 19x$ , for all real numbers x and y. The value of x for which $f(x, 2x) = 27$ , is